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Theorem tpostpos2 8271
Description: Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
tpostpos2 ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹)

Proof of Theorem tpostpos2
StepHypRef Expression
1 tpostpos 8270 . 2 tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V))
2 relrelss 6295 . . . 4 ((Rel 𝐹 ∧ Rel dom 𝐹) ↔ 𝐹 ⊆ ((V × V) × V))
3 ssun1 4188 . . . . . 6 (V × V) ⊆ ((V × V) ∪ {∅})
4 xpss1 5708 . . . . . 6 ((V × V) ⊆ ((V × V) ∪ {∅}) → ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V))
53, 4ax-mp 5 . . . . 5 ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)
6 sstr 4004 . . . . 5 ((𝐹 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V))
75, 6mpan2 691 . . . 4 (𝐹 ⊆ ((V × V) × V) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V))
82, 7sylbi 217 . . 3 ((Rel 𝐹 ∧ Rel dom 𝐹) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V))
9 dfss2 3981 . . 3 (𝐹 ⊆ (((V × V) ∪ {∅}) × V) ↔ (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹)
108, 9sylib 218 . 2 ((Rel 𝐹 ∧ Rel dom 𝐹) → (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹)
111, 10eqtrid 2787 1 ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  Vcvv 3478  cun 3961  cin 3962  wss 3963  c0 4339  {csn 4631   × cxp 5687  dom cdm 5689  Rel wrel 5694  tpos ctpos 8249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-tpos 8250
This theorem is referenced by:  2oppchomf  17771  mattpostpos  22476  opprabs  33490
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